Logic/Model Theory Seminar | Daniel Max Hoffmann | Amenable tame flows: Theorems motivated by model-theoretic dynamics

There is an interesting interplay between model theory and topological dynamics. Newelski suggested using the machinery of topological dynamics to study definable groups and the related dynamics on spaces of types. For example, he proposed a conjecture stating that, under suitable model-theoretic assumptions, the Ellis group of a flow should be isomorphic to the compact group (G/G^{00}). This conjecture was confirmed for definably amenable groups in NIP by Chernikov and Simon (2018), but was also refuted in NIP for SL(2,R) by Gismatullin, Penazzi and Pillay (2015). Krupiński and Pillay formulated a weaker version, the so-called revised Newelski conjecture, stating that, under NIP, the Ellis group associated with definable group dynamics has a Hausdorff (\tau)-topology. This revised conjecture was confirmed by Chernikov, Gannon and Krupiński for definable group dynamics in a metrizable setting. Basso and Zucker then extended this result to the non-metrizable setting via forcing; in fact, they generalized some parts of Glasner’s structural results.

Krupiński and I observed that an abstract variant of Newelski’s original conjecture follows from Glasner’s results for amenable, tame, minimal and metrizable flows. Our goal was then to remove the metrizability assumption, as had been done in the case of the revised conjecture. This led to results on the amenability of the Ellis semigroup and on the measurable structure of amenable minimal tame flows. These results will be presented in two papers.

The first is already available on arXiv: https://arxiv.org/abs/2607.24381.

MC 5403